Question 1140601
Derivative of {{{y = 2^(pi*x) }}}


{{{(d/dx) (2^(pi*x) )}}}


Apply exponent rule : {{{a^b=e^(b*ln (a))}}}:


{{{2^(pi*x) =e^(b*ln (a))}}}


={{{(d/dx) (e^(b*ln (a)))}}}


Apply the chain rule : {{{df(u)/dx= (df/du)*(du/dx)}}}

where {{{f=e^u}}}, {{{u=pi*x*ln(2)}}}


={{{(d/du)(e^u)*(d/dx)(pi*x*ln(2))}}}

={{{e^u*pi*ln(2)}}}


Substitute back :{{{u= pi *x*ln (2)}}}:


={{{e^(pi *x*ln (2))*pi*ln(2)}}}


since {{{e^(pi *x*ln (2))= 2^(pi*x)}}}
 

={{{2^(pi*x) *pi*ln(2)}}}


so, {{{(d/dx) (2^(pi*x) )=2^(pi*x) *pi*ln(2)}}}